Maybe you are interested in this little PoC I made a while ago which instead sends a one-time-auth code using Matrix: https://loginwithmatrix.tiktalk.space/
It does prove that you own the specific matrix id you are logging in with to the website. It can for example be used as an alternative everywhere where "Sign in with Google" is used, so I don't understand why it should be useless for applications.
Aren't you supposed to use "Sign in with Google" or which ever service to prove to the app that you own your unique id (e.g. email)? If you want Google APIs, then that's a completely different scope here.